Incomparability in parallel computation

We consider the relative power of concurrentwrite PRAMs when the number of processors (and input variables) is fixed at n, and infinite shared memory is allowed. Several different models (COMMON, ARBITRARY, PRIORITY) have been used for algorithm design in the literature; these models differ in their method of write-conflict resolution. Recent work in separating these models ([FRW1,2,3], [LY]) has relied on further restrictions (limiting the size of memory or the power of processors); the only unrestricted results known concern the element distinctness problem ([FMW], [RSSW]). In this paper we contribute further unrestricted results. We consider the COLLISION model, a natural generalization of the Ethernet ([G]). Our main result is a lower bound of Ω(logloglogn) steps on COLLISION for a problem that can be done in O(1) steps on ARBITRARY. We use this result, together with a reduction performed by means of Ramsey's Theorem, to show that the powers of COMMON and COLLISION are incomparable. We also introduce a new and natural model, TOLERANT, and show that it is strictly less powerful than COLLISION and incomparable with COMMON. The proofs involved use combinatorial arguments, including Turán's Theorem for graphs and the Erdös-Rado intersecting set theorem.